helm/software/matita/library/dama/nat_ordered_set.ma

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  14
  15 include "nat/compare.ma".
  16 include "dama/bishop_set.ma".
  17
  18 definition nat_excess : nat → nat → CProp ≝ λn,m. m<n.
  19
  20 lemma nat_elim2:
  21   ∀R:nat → nat → CProp.
  22   (∀ n:nat. R O n) → (∀n:nat. R (S n) O) → (∀n,m:nat. R n m → R (S n) (S m)) →
  23     ∀n,m:nat. R n m.
  24 intros 5;elim n; [apply H]
  25 cases m;[ apply H1| apply H2; apply H3 ]
  26 qed.
  27
  28 alias symbol "lt" = "natural 'less than'".
  29 lemma nat_discriminable: ∀x,y:nat.x < y ∨ x = y ∨ y < x.
  30 intros (x y); apply (nat_elim2 ???? x y);
  31 [1: intro;left;cases n; [right;reflexivity] left; apply lt_O_S;
  32 |2: intro;right;apply lt_O_S;
  33 |3: intros; cases H;
  34     [1: cases H1; [left; left; apply le_S_S; assumption]
  35         left;right;rewrite > H2; reflexivity;
  36     |2: right;apply le_S_S; assumption]]
  37 qed.
  38
  39 lemma nat_excess_cotransitive: cotransitive ? nat_excess.
  40 intros 3 (x y z); unfold nat_excess; simplify; intros;
  41 cases (nat_discriminable x z); [2: left; assumption] cases H1; clear H1;
  42 [1: right; apply (trans_lt ??? H H2);
  43 |2: right; rewrite < H2; assumption;]
  44 qed.
  45
  46 lemma nat_ordered_set : ordered_set.
  47 letin hos ≝ (mk_half_ordered_set nat (λT,R:Type.λf:T→T→R.f) ? nat_excess ? nat_excess_cotransitive);
  48 [ intros; left; intros; reflexivity;
  49 | intro x; intro H; apply (not_le_Sn_n ? H);]
  50 constructor 1;  apply hos;
  51 qed.
  52
  53 interpretation "ordered set N" 'N = nat_ordered_set.
  54
  55 alias id "le" = "cic:/matita/nat/orders/le.ind#xpointer(1/1)".
  56 lemma os_le_to_nat_le:
  57   ∀a,b:nat_ordered_set.a ≤ b → le a b.
  58 intros; normalize in H; apply (not_lt_to_le b a H);
  59 qed.
  60
  61 lemma nat_le_to_os_le:
  62   ∀a,b:nat_ordered_set.le a b → a ≤ b.
  63 intros 3; apply (le_to_not_lt a b);assumption;
  64 qed.
  65